Fractal Geometry in Mesoamerica

This information has been digitized for use in the Ethnomathematics Digital Library (EDL), a program of Pacific Resources for Education and Learning (PREL). The EDL is sponsored by the National Science Foundation as a part of the National STEM Digital Library (www.nsdl.org).

FRACTAL GEOMETRY IN MESOAMERICA

by Gerardo Burkle Elizondo

Universidad Autonoma de Zacatecas Zacatecas, Mexico

© Symmetry Foundation. Digitized 2004 by permission of publisher. Elizondo, G. B. (2001). Fractal geometry in Mesoamerica [Special issue of Symmetry: Culture and Science]. Symmetry in Ethnomathematics, 12(1-2), 201-214. Budapest, Hungary: International Symmetry Foundation. This product was funded by the National Science Foundation (NSF) as a component of the National Science, Technology, Engineering, and Mathematics Education Digital Library (NSDL), award number DUE0121749. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of NSF.

Symmetry: Culture and ScienceVol. 12, Nos. 1-2, 201-214, 2001

FRACTAL GEOMETRY IN MESOAMERICA

Gerardo Burkle Elizondo

Doctorado En Historia Del Centro Interinstitucional De Investigaciones En Artes Y Humanidades De LaUniversidad Autonoma De Zacatecas.

Address: Colina de San Antonio #105, Las Colinas, CP 98064, Zacatecas, Zac. México. E-mail:[email protected].

Abstract: Fractal Dimension in different groups of mesoamerican artistic andarchitectural works was quantified in 106 structures. To make the analysis we used the“Benoit” program in order to calculate “Box Dimension”, “Information Dimension”,“Mass Dimension” and “Y” value about “Fractal Dimension”. We found a generalaverage of 1.92 for fractal dimension with values by groups in a range between 1.883and 2.038. These results are in a very good agreement with the fact that the values ofcomplex fractals are usually situated between 1.5 and 2.0. Fractal dimension shows theefficiency with which an object fills the space that contains it, and is quantificated likethe morphology of their complexity.

Scaling properties exponent were proved, and specifically the fractal dimensionobtained looks like a possible pattern. We think that part of the simple rules that couldexplain these complex dynamics findings is the fact that the artists and architects inMesoamerica used mathematics in their work. The finding of “proportional systems”and “golden units” to measure in modular forms, like scaling rectangles and othertechniques to get harmonic and constant units in the case of Mesoamerica, had beenproved by some authors.

INTRODUCTION

The study of geometric analysis and mathematics that were used in art composition andconstruction in Mesoamerica is not yet clearly established.

202 G. BURKLE ELIZONDO

Some approaches have been developed, and in general agree with the fact that there wasa basic form that fundamentally consisted in the use of “golden unit measurements”.

In her book, Margarita Martínez del Sobral (2000) speaks, as did before Beatríz de laFuente about the Olmec sculpture (1984), about the existence of proportional systemsconsisting of rectangular standardized modules of standardized measurements. Theserectangles were in fact golden and harmonic units.

She found some constants in this way, basically different golden rectangle measures,and a developing in a spiral harmonic method related to growing rectangles.

Therefore the figures appear like proportions in a module that can be geometricallyanalyzed. The rectangles can also move and grow in a spiral way to get, with therotations, new and bigger rectangles but in proportion with its golden basic initial unit.

Also she shows in her book the “Fibonacci serie” and finds many turn angles that wereused in Mesoamerica in art and in architecture and urbanism.

Both, De la Fuente and Martínez had proven that in the prehispanic world a system likethis was used constantly by many cultures over the times, as a well developed geometricscheme by scribes (Tlacuilos), artists, sculptors and architects making of this astandardized technique and a universal tool in composition, and that these rules weretransmitted like a tradition from one generation to another.

The fact that the use of mathematics was a conscious planning device is not discussednow, but we need to try to understand better about the techniques that were used tocalculate, design and create aesthetic proportions, in order to know what the evolutionof art and history really was.

HYPOTHESES

Mesoamerican astronomers developed, as is well known, a deep knowledge about themovement of stars and planets. They in fact made a connection between the cosmicgeometric scheme that learned from the sky, and their own lives. To them all wererelated to sacredness and to the gods: their art, architecture and even the places and formof construction and the orientation of the cities. The part of archaeology that studies thismatter is named “archaeoastronomy”.

FRACTAL GEOMETRY AT MESOAMERICA 203

The Maya mathematical system was a vigesimal one. Eric Thompson (1950, p. 143)found the “uinic” like a standard measure whose value was 20 units in a cord-lengthused to measure the land.

Before him the linguist Daniel Brinton (1890, pp. 433-451) reported that he found“words for measurings” in nahuatl and maya-yucateco languages that correspond tosome of the dimensions of the human body, for example the arm and the foot andothers.

Some authorities like V. Garth Norman had found a very probable mesoamericancultural marker in a “codo” equivalent, whose measurement is 49.5 centimeters or 19.49inch, and that this and their multiples and divisions were used, for example, inTheotihuacan and in many parts of the Maya area.

The engineer Hugh Harleston (1974) who made measurements of the Theotihuacanpyramids finally also found the smallest fraction represented by the number 19 andother standards very close to a meter.

In a previous paper we reported the finding of Fractal Dimension in manymesoamerican art and architectural works (Burkle 2001). Now we collected a greaternumber of structures, especially a big and representative number of pyramids toincrease the value.

From the foregoing evidence, we think that the use in this field of “fractal geometry”would make sense in two ways: first we hypostatize that mesoamerican art, sculpture,aesthetics and architecture, with particular attention to the mayas, are “fractals” or atleast have fractal dimension in their basic form. This could mean that the designs, theprojects and their conceptions are easier to understand in a base of complex patternswith up or down scale properties, that are characterized by a fractal exponents withparticular magnitudes. We found these values between 1.883 and 2.038 in the averagesof ten groups that we submit to analysis.

The second point is that we think that probably the mayas knew and used mathematicsthat today science calls “non-linear”, “scalant”, “boundaries”, “Fibonacci series”,“extrange attractors”, “self-affinity”, “self-similarity”, “fractals”, concepts that belongto “chaos theory”.


Confirmation about the discovery of a standard unit requires different ways of study andprecise and consistent use checking, and we think that the “fractal and chaos theory”application could help to increase the understanding of the real dimension and grade ofmesoamerican cultural development.

METHODS

Today fractal analysis is a universal tool that had been applied to many sciences. Itseems that no previous study had been done about its application to understand ancientmathematics, especially those that were used in México during the prehispanic period.

We know that they were very good mathematicians, and in fact our findings in thisstudy confirm this, especially because in our fractal analysis we found a very closegeometric pattern has is shown in the results.

The study was divided in 10 groups:

1. Walls sculptures of Palenque temples

2. Maya stelas of different places

3. Maya hieroglyphs from Palenque

4. Pyramids and temples from various places of Mesoamerica

5. Different calendars in stones and codex (Tonalamatl)

6. Various codex pages, specially from the Dresde one

7. Murals of different parts of Mesoamerica

8. Great stone monuments like the Aztec calendar, Coatlicue, a Tula giant, stela D fromCopán and a colossal head from La Venta

9. Astronomic stones and

10. Figures in ceramic from the Maya culture.

The ten groups give a total of 106 figures that were analysed and that are described inTable 1. Basically the images correspond first to Maya culture, developed at Chiapasand Yucatán in México, Tikal at Guatemala and Copán Honduras, and to the Aztec orMexica cultures developed at Mexican Central High-plains, and some to the Toltec andOlmec cultures from México.


God Coyoxauqui of the Aztec culture at México

Temple Pyramid of Tikal of the Maya culture at Guatemala


Disc metal plate of the Cenote Sagrado of Chichen Itzá, of the Maya culture at Yucatán, México

Codex Fejérváry-Mayer page 1, Mixteca culture from México


Temple of the Sun Panel at Palenque Chiapas, México

To make the analysis these figures were first scanned and saved as bitmap files.Thereafter the images were analysed with the “Benoit Program version 1.3” for FractalAnalysis System. Then we calculated three main characteristics about the fractaldimension of each one: its Box Dimension, Information Dimension, and the MassDimension with the “Y” value in each case with their respective standard errors, andintercepts on log-log plots graphs.

The selection of the structures and images were non-random.

Table 1: Art and Architecture Analysed Groups

I. PALENQUE AND OTHER TABLETS (Milbrath 1999).

II. MAYA STELAS AND OTHERS (Aveni 1997).

III. MAYA HIEROGLYPHS (Milbrath 1999).

IV. PYRAMIDS AND TEMPLES OF MESOAMERICA (Marquina 1964).


V. CALENDARS (TONALAMATL) AND ASTRONOMIC STONES(Thompson 1950).

VI. CODEX PAGES (Harleston 1974).

VII. MURALS OF MESOAMERICA CITYES (Aveni 1997).

VIII. GREAT STONE MONUMENTS (Tyler 1995).

IX. ASTRONOMIC STONES (Broda Prucha 1991).

X. CERAMIC VASEES AND OTHERS (Schele and Grube 1997).

EXPLANATION ABOUT THE ANALYSIS(BENOIT FRACTAL ANALYSIS SYSTEM)

To get our data we used the program (TruSoft Int’l Inc. Benoit, version 1.3: FractalAnalysis System. 20437th Ave. No. 133, St. Petersburg, FL 33704, USA) we hadmentioned before. This analysis system permits to get different kinds of fractalcalculations (fractal dimensions).

We chose three characteristics to be studied because it seems to be specifically useful tothe understanding of the dynamics we were studying. These three methods are:

a) Box Dimension;

b) Information Dimension and

c) Mass Dimension. In each case we get too the “Y” value.

The three methods belong to self-similar pattern “x” analysis, and can be betterunderstood if we pay attention to some of the relevant aspects: first, if we begin usingsome important aspects of chaos theory, actually in this case the use of concepts like“order” can help us to look at the fundamental tasks of this work and, second: fractalsare in fact “the geometry of chaos”, but a special kind of chaos in which order emergeswith laws governing the description of the system.

Starting from the properties of a “fractal” as the principles that make reference to itsown characteristics like:

1. The value of its dimension, that is between 1 and 2 depending on the occupied spaceand the dynamic (Hausdorff dimension).


2. Self similarity is another characteristic that means that the form does not changewhen the scale does, but that becomes more and more complex (complexity) in the non-linear fractals, especially the 2 nearest values.

3. The self-similarity in fact mathematically means a potential relation between the sizeand the measurement or scale.

4. Its dimension is fractional or fractal and non-Euclidean.

5. They have an infinite longitude. The data set has scale invariance characteristics andexhibits a fractional motion with statistical properties in a curve with a scaling factorbetween 1 and 2 (Hurst exponent). This scale invariance can be tested studying the setin many length scales, which can be done with the “Benoit Fractal Analysis System”that we used to get the information about the figures to verify first if they were or werenot possible fractals, their fractal dimensions and get the exponent and try to see if thisis a pattern that can be may be recognized like a characteristic marker in the results.

About “Dimensions” - Box Dimension Estimation Method Interface shows “the numberof boxes of linear size x necessary to cover a data set of points distributed in a two-dimensional plane”. In the “Benoit” program each box and grid rotates 90 degrees sizethrough dividing to get the minimum value of the x dimension.

Information Dimension Estimation Method Interface “assigns weights to the boxes”.These boxes that contain more points or those that contains less, are analyzed from thepoint of view of their masses contained in “each box” (information entropy).

Mass Dimension Estimation Method Interface implicates the number of points inside acircle around the data set in a two-dimensional plane to define the mass; this can bemeasured in “circles of increasing radius”.

These three methods will reveal first if the set have fractal dimension, and itscomplexity degree.

About “Y” - Fractal dimension is expressed like the slope in the graph. The factorrepresents the lagunarity or empty spaces distribution related to those non-empty, andthe exponent represents the fractal dimension.

The graphs a, b and c correspond to Fractal Analysis of the Glyph 14 from PalenquePalace tablet, respectively Box Dimension (Db), Information Dimension (Di) and MassDimention (Dm).


RESULTS

We could summarize our findings in the following order:

1. In all the 106 mesoamerican artistic and architectural images that were analysed wefound fractal dimension, and all were “complex”, with a Df between 1.8 and 2.0.

2. We find that this is a very close range between the different dimensions of the tengroups of elements that were analysed. This could mean that this may be a fractalmarker for Mesoamerica measurement (1.92) with the method we used.

3. We find a pattern present in the average of each group, specifically in the morecomplex groups. This probably reflects the use of golden units we had mentioned beforethat had been reported by other authors.

4. In fact this method of study, the fractals and chaos theory applied to art, history andancient scientific culture seems to be a good and interesting tool of complexity theoryfor analysis to still studying.

5. We can see about the range we spoke about, that it is at the highest level ofcomplexity. This could be understood as order emerging from the edge of chaoscritically, which in fact define the concept of “complexity”.

In Table 2 we can see the Box Dimension (Db), Information Dimension (Di) and theMass Dimension (Dm) averages by groups of study.

GroupNo. Db ± SD Di ± SD Dm ± SD

I 1.918±0.010 1.932±0.002 2.018±0.111II 1.923±0.007 1.940±0.001 1.887±0.060III 1.910±0.008 1.903±0.003 2.036±0.088IV 1.926±0.011 1.933±0.006 1.975±0.010V 1.921±0.008 1.926±0.002 1.937±0.051VI 1.918±0.009 1.924±0.003 2.038±0.269VII 1.919±0.006 1.929±0.002 1.964±0.058VIII 1.917±0.009 1.914±0.003 1.954±0.053IX 1.900±0.006 1.877±0.003 1.975±0.047X 1.883±0.013 1.888±0.003 1.966±0.214

Table 2


The images of the next page belongs to some of the structures that were studied such asa Jaguar sculpture from Teotihuacan México with Db = 1.917±0.003, Di = 1.915±0.001and Dm = 1.934±0.017.

The next image belongs to the Stela F from Quirigua Guatemala with Db == 1.940±0.011, Di = 1.930±0.009 and Dm = 2.142±0.276.

The analysis of the disc of Xochimilco from the Aztec culture at México shows a Db == 1.9008±0.0075, Di = 1.874±0.003 and a Dm = 1.994±0.048.

These are some examples about the results that we got in our study.

GENERAL CONCLUSIONS AND COMMENTS

It is very interesting to discover that for all the 106 cases that were studied ofmesoamerican culture images, all of them have Fractal Dimension. Some of thesebelong to those kinds of fractals or fractal geometry that can be discovered at first sightin its proportions and its dimensions, and are in general figures that imitate the nature’saesthetics in some way.

Others have greater complexity, maybe because its self-similarity or self-affinity isstatistic and anisotropic just as real objects are.

In both cases that we found out there could be some kind of metapatterns that need to beunderstood better.

It is necessary to make more studies to get conclusions, but with the information wegathered, we think that there undoubtedly existed a mathematical system and a deepgeometrical development in mesoamerican art and architecture, and that they usedpatterns and “golden units”.

We also think that these wise men used complex mathematics to make their works,maybe non-linear and chaotic, trying to imitate the movements in the sky and thecomplexity of the nature’s phenomena to establish some kind of similitude from thisside, and the world were the gods live, trying then to understand in a holy way theirdesigns and random wishes.


They needed to get the best possible information about the world, the universe, theimplied order that exists between them and the sacred cosmos that governed theiruncertain lives.

Dr. Joana Broda (1991, p. XII) says that in the archaic past, the science was historicallydeterminate in civilizations that, like now, assumed science “like a part of a socialwhole”.

The use of fractals and chaos theory as a tool to study history for certain will open newdoors of understanding between social sciences and things that we perhaps can notimagine now.

In his book “Metapatterns”, Tyler Volk (1995, p. 206) asks the following metaphoricidea: “…Are boarding crossings in space the inevitable tool for portraying time’s breaksin diagrams and myths?”.

REFERENCESAveni, A. F. (1997) Observadores del Cielo en el México Antiguo, Fondo de Cultura Económica.Briggs, J. (1992) Fractals. The Patterns of Chaos, Touchstone, Simon and Schuster.Broda Prucha, J. (1991) Arqueoastronomía y Etnoastronomía en Mesoamérica, Universidad Nacional

Autónoma de México.Burkle Elizondo, G. (2001) Valdéz Cepeda Ricardo. Do the Mesoamerican Artistic and Architectural Works

Have Fractal Dimension?, In: Miroslav M. Novak, ed., Emergent Nature. Patterns, Growth and Scalingin the Sciences, pp. 431-432. World Scientific.

Closs, M. P. (1990) Native American Mathematics, University of Texas Press.Cowan, G. A. et al. (1999) Complexity. Metaphors, Models, and Reality, Perseus Books.Daniel, B. G. (1890) The Lineal Measures of the Semi-civilized Nations of Mexico and Central América,

Essays of an Americanist, Poter and Coates.De La Fuente Beatríz y cols (1999) Pintura Mural Prehispánica, Jaca Book/CONACULTA.Fuente, B. de la (1984) Los Hombres de Piedra, Escultura Olmeca, Universidad Nacional Autónoma de México.Garcés Contreras Guillermo (1995) Pensamiento Matemático y Astronómico en el México Precolombino,

Instituto Politécnico Nacional.Harleston, H. Jr. (1974) A Mathematical Analysis of Theotihuacan, XLI International Congress of

Americanists, October 3.Kauffman, S. A. (1993) The Origins of Order, Oxford University Press.Mandelbrot, Benoit (1997) La Geometría Fractal de la Naturaleza, Tusquets Editores.Mangino Tazzer Alejandro (1996) Arquitectura Mesoamericana. Relaciones Espaciales, Trillas.Marquina, I. (1964) Arquitectura Prehispánica. Instituto Nacional de Antropología e Historia, Secretaría de

Educación Pública. México.Milbrath, S. (1999) Star Gods of the Maya. Astronomy in Art, Folklore, and Calendars, University of Texas Press.Peitgen Heinz, O. et al. (1992) Chaos and Fractals. New Frontiers of Science, Springer Verlag.Schele, L. and Freidel, D. (1999) Una Selva de Reyes. La Asombrosa Historia de los Antiguos Mayas, Fondo

de Cultura Económica.Schele, L. and Grube, N. (1997) The Proceedings of the Maya Hieroglyphic Workshop, The Dresde Codex,

March 8-9.Schroeder, M. (1999) Fractals, Chaos, Power Laws, W. H. Freman and Company.Sobral Margarita, M. del (2000) Geometría Mesoamericana, Fondo de Cultura Económica.Thompson, J. E. S. (1950) Maya Hieroglyphic Writing. Carnegie Institution of Washington, publication 589.Tyler, V. (1995) Metapatterns. Across Space, Time, and Mind, Columbia University Press.

Fractal Geometry in Mesoamerica

Documents

Transcript of Fractal Geometry in Mesoamerica